Make an object only visible from a certain angle?
The answer is using a Light Path node, just like Todd just said, but I'm using a different approach and a different node factor, since according to attached images, this has to happen while both mouths would be visible from a 3/4 angle:
1- Create a plane to go directly in the center of your character, The cube in the image here represents your character's head, the two spheres represent the mouth.
2- Assign a simple transparent material to the plane.
3- Mix a transparent shader to the material of the mouths (the spheres in this example), add a Light Path node, Add node>Input>Light Path, use the Transparent depth as a factor. This will make the object that this material assigned to, disappears when it falls behind an object with a transparent material, which is the plane here. Check the image below.
That's it check the animated GIF below to see how this will work, no need to say that from a front view, both of the mouths will show, so you can extrude the far edge of the transparent plane to the left or right to solve that issue.